Axiom of finite choice

In mathematics, the axiom of finite choice is a weak version of the axiom of choice which asserts that if ${\displaystyle (S_{\alpha })_{\alpha \in A}}$ is a family of non-empty finite sets, then

${\displaystyle \prod _{\alpha \in A}S_{\alpha }\neq \emptyset }$ (set-theoretic product).^[1]: 14

If every set can be linearly ordered, the axiom of finite choice follows.^[1]: 17

Applications

An important application is that when ${\displaystyle (\Omega ,2^{\Omega },\nu )}$ is a measure space where ${\displaystyle \nu }$ is the counting measure and ${\displaystyle f:\Omega \to \mathbb {R} }$ is a function such that

${\displaystyle \int _{\Omega }|f|d\nu <\infty }$,

then ${\displaystyle f(\omega )\neq 0}$ for at most countably many ${\displaystyle \omega \in \Omega }$.